Equation For The Pseudo-Inverse OLS Estimator - Data Values for a Regression Problem - Statistics Assessment Answer

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Statistics Assessment Answer

Task: Assignment Question It is to be determined whether the fruit yield of tomato plants depends on the amount used of four additives for the soil. The regression model shown here will be applied: Y = ? + ?1 X1 + ?2 X2 + ?3 X3 + ?4 X4 Where • Y is the response variable of the weight of fruit yielded, • Xi is the variable of the amount, in appropriate units, of additive i for i = 1 to 4 and • The numbers ?i are the corresponding coefficients for each additive. Answer the following questions. 1. Write out the elements of the matrices X^T X and X^T Y in the equation for the pseudo-inverse OLS estimator for parameter vector b = (?, ?i ) T. The elements of these matrices should be written in terms of the Y i and Xi, j. 2. Use these results to calculate the OLS estimator b for the data given at the end of the assignment document. You can calculate b by finding the inverse of the appropriate matrix by hand (not recommended!) in the OLS equation as it stands, by writing a piece of code to find the inverse or by using Cramer’s rule on the matrix problem (X^TX)^b = (X^T)Y. 3. Set up the Null and alternative Hypothesis and other elements of the test to investigate whether the last two variables are required to‘explain’ Y. Your test statistic will be 4. Repeat this work to investigate whether all four variables are required to ‘explain’ Y.

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